Abstract

In this paper, we study the existence and uniqueness of solutions for the boundary value problem of fractional difference equations

{−Δνy(t)=f(t+ν−1,y(t+ν−1)),y(ν−3)=0,Δy(ν−3)=0,y(ν+b)=g(y)

and

{−Δνy(t)=λf(t+ν−1,y(t+ν−1)),y(ν−3)=0,Δy(ν−3)=0,y(ν+b)=g(y),

respectively, where t=1,2,…,b, 2<ν≤3, f:{ν−1,…,ν+b}×R→R is a continuous function and g∈C([ν−3,ν+b]Zν−3,R) is a continuous functional. We prove the existence and uniqueness of a solution to the first problem by the contraction mapping theorem and the Brouwer theorem. Moreover, we present the existence and nonexistence of a solution to the second problem in terms of the parameter λ by the properties of the Green function and the Guo-Krasnosel’skii theorem. Finally, we present some examples to illustrate the main results.

MSC:34A08, 34B18, 39A12.

Keywords

1 Introduction

In recent years, fractional differential equations have been of great interest. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry and engineering. Mathematicians have employed this fractional calculus in recent years to model and solve a variety of applied problems. Indeed, as Podlubny outlines in [1], fractional calculus aids significantly in the fields of viscoelasticity, capacitor theory, electrical circuits, electro-analytical chemistry, neurology, diffusion, control theory and statistics.

The continuous fractional calculus has developed greatly in the last decades. Some of the recent progress in the continuous fractional calculus includes the paper [2], in which the authors explored a continuous fractional boundary value problem of conjugate type using cone theory, they then deduced the existence of one or more positive solutions. Of particular interest with regard to the present paper is the recent work by Benchohra et al. [3]. In that paper, the authors considered a continuous fractional differential equation with nonlocal conditions. Other recent work in the direction of those articles may be found, for example, in [4–12].

In recent years, a number of papers on the discrete fractional calculus have appeared, such as [13–30], which has helped to build up some of the basic theory of this area. For example, Atici and Eloe discussed the properties of the generalized falling function, a corresponding power rule for fractional delta-operators and the commutativity of fractional sums in [13]. They presented in [13] more rules for composing fractional sums and differences. Goodrich studied a two-point fractional boundary value problem in [16], which gave the existence results for a certain two-point boundary value problem of right-focal type for a fractional difference equation. At the same time, a number of papers appeared, and these began to build up the theoretical foundations of the discrete fractional calculus. For example, a recent paper by Atici and Eloe [14] explored some of the theories of a conjugate discrete fractional boundary value problem. Discrete fractional initial value problems were considered in a paper by Atici and Eloe [15].

Atici and Eloe in [14] considered a two-point boundary value problem for the finite fractional difference equation

−Δνy(t)=f(t+ν−1,y(t+ν−1)),t=1,2,…,b+1,y(ν−2)=0,y(ν+b+1)=0,

where 1<ν≤2 is a real number, b≥2 is an integer and f:[ν,ν+b]Nν−1×R→R is continuous. They analyzed the corresponding Green function, provided an application and obtained sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.

Goodrich in [18] considered a discrete fractional boundary value problem of the form

−Δνy(t)=f(t+ν−1,y(t+ν−1)),y(ν−2)=0,y(ν+b)=g(y),

where t∈[0,b]N0:={0,1,…,b}, f:[ν−2,ν−1,…,ν+b−1]Nν−2×R→R is a continuous function, g∈C([ν−2,ν+b]Zν−3,R) is a given functional, and 1<ν≤2. He established the existence and uniqueness of a solution to this problem by the contraction mapping theorem, the Brouwer fixed point theorem and the Guo-Krasnosel’skii fixed point theorem.

Although the boundary value problem of fractional difference equations has been studied by several authors, the present works are almost all concerned with 1<μ≤2, very little is known in the literature about a fractional difference equation with 2<μ≤3.

Motivated by all the works above, in this paper, we first aim to study the following boundary value problem:

where t=0,1,2,…,b, 2<ν≤3, f:{ν−1,…,ν+b−1}Nν−1×R→R is continuous and g∈C([ν−3,ν+b]Zν−3,R) is a continuous functional, λ is a positive parameter. We establish some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem by considering the eigenvalue intervals of the nonlinear fractional differential equation with boundary conditions.

The plan of this paper is as follows. We first give the form of solutions of problem (1.1), second we prove the existence and uniqueness of a solution to problem (1.1) by the contraction mapping theorem and the Brouwer theorem, and then the eigenvalue intervals for the boundary value problem of nonlinear fractional difference equation (1.2) are considered by the properties of the Green function and the Guo-Krasnosel’skii fixed point theorem on cones. Finally we present some examples to illustrate the main results.

2 Preliminaries

For the convenience of the readers, we first present some useful definitions and fundamental facts of fractional calculus theory, which can be found in [13, 14].

We define tν̲:=Γ(t+1)Γ(t+1−ν) for any t and ν, for which the right-hand side is defined. We also appeal to the convention that if t+1−ν is a pole of the gamma function and t+1 is not a pole, then tν̲=0.

The proof of this theorem is similar to that of Theorem 3.2 in [14]. Hence, we omit the proof here.

3 Existence and uniqueness of solution

In this section, we wish to show that under certain conditions, problem (1.1) has at least one solution. We know that problem (1.1) can be recast as an equivalent summation equation. It follows from Lemma 2.4 that y is a solution of (1.1) if and only if y is a fixed point of the operator T:Rb+4→R, where

Then condition (3.2) holds. We find that (1.1) has a unique solution, which completes the proof of the theorem. □

By weakening the conditions imposed on f(t,y) and g(y), we can still obtain the existence of a solution to (1.1). We apply the Brouwer theorem to accomplish this.

Theorem 3.2Suppose that there exists a constantK>0such thatf(t,y)satisfies the inequality

max(t,y)∈[ν−3,ν+b]Zν−3×[−K,K]|f(t,y)|≤K2Γ(ν+b+1)Γ(ν+1)Γ(b+1)+1

(3.7)

andg(y)satisfies the inequality

maxy∈[−K,K]|g(y)|≤K2Γ(ν+b+1)Γ(ν+1)Γ(b+1)+1.

(3.8)

Then (1.1) has at least one solutiony0satisfying|y0(t)|≤Kfor allt∈[ν−3,ν+b]Zν−3.

Proof Consider the Banach space B:={y∈Rb+4:∥y∥≤K}. T is defined as (3.1). It is obvious that T is a continuous operator. Therefore, our main objective is to show that T:B→B. That is, whenever ∥y∥≤K, it follows that ∥Ty∥≤K. Once this is established, we use the Brouwer theorem to deduce the conclusion.

Assume that inequalities (3.7) and (3.8) hold for given f and g. For convenience, we let

On the one hand, from Lemma 2.1 we know tν−1̲ is increasing in t, thus we have

∑s=0b(ν+b−s−1)ν−1̲=[−1ν(ν+b−s)ν̲]s=0b+1=Γ(ν+b+1)νΓ(b+1).

(3.12)

On the other hand,

tν−1̲(ν+b)ν−1≤1.

(3.13)

Inserting (3.11)-(3.13) into (3.10), we can obtain

∥Ty∥≤Φ[2Γ(ν+b+1)Γ(ν+1)Γ(b+1)]+Φ=Φ[2Γ(ν+b+1)Γ(ν+1)Γ(b+1)+1].

(3.14)

By substituting (3.9) into (3.14), we have

∥Ty∥≤Φ[2Γ(ν+b+1)Γ(ν+1)Γ(b+1)+1]=K.

(3.15)

Thus, from (3.15) we deduce that T:B→B. Consequently, it follows at once by the Brouwer theorem that there exists a fixed point of the map T, say Ty0=y0 with y0∈B. So, this function y0 is a solution of (1.1) and y0 satisfies the bound |y0(t)|≤K for each t∈[ν−3,ν+b]Zν−3. And this completes the proof of the theorem. □

4 Existence of a positive solution

In this section, we show the existence of positive solutions for boundary value problem (1.2).

Letℬbe a Banach space, and letK⊆Bbe a cone. Assume thatΩ1andΩ2are two bounded open subsets contained inℬsuch that0∈Ω1andΩ¯1⊆Ω2. Assume further thatT:K∩(Ω¯2∖Ω1)→Kis a completely continuous operator. If either

(1)

∥Ty∥≤∥y∥fory∈K∩∂Ω1and∥Ty∥≥∥y∥fory∈K∩∂Ω2, or

(2)

∥Ty∥≥∥y∥fory∈K∩∂Ω1and∥Ty∥≤∥y∥fory∈K∩∂Ω2,

thenThas at least one fixed point inK∩(Ω¯2∖Ω1).

Define the Banach space ℬ by

B={y:[ν−3,ν+b]Zν−3→R:y(ν−2)=y(ν−3)=0,y(ν+b)=g(y)},

with the norm ∥y∥=max{|y(t)|,t∈[ν−3,ν+b]Zν−3}.

For tν−1̲ is increasing, we get maxt∈[ν−3,ν+b]Zν−3tν−1̲(ν+b)ν−1̲=1. Thus, there exists a positive constant γ0 such that

Hence by Theorem 5.2 we deduce that (6.6) has no positive solution for λ>λ0.

7 Conclusion

This paper is an extension of [14] and [18]. The main contributions of this paper include:

The existence and uniqueness of a solution to a class of boundary value problems for a fractional difference equation with 3<α≤4 are studied by the contraction mapping theorem.

The existence of a solution to a class of boundary value problems for a fractional difference equation with 3<α≤4 is studied by the Brouwer fixed point theorem.

The eigenvalue intervals of a boundary value problem for a class nonlinear fractional difference equations with 3<α≤4 are investigated by the Guo-Krasnosel’skii fixed point theorem.

The nonexistence of a positive solution boundary value problem for a class nonlinear fractional difference equations with 3<α≤4 is considered in terms of parameter.

In contrast to [14] and [18], the similarities and differences are as follows:

The methods used to prove the existence results are standard and the same; however, their exposition in the framework of problems (1.1) and (1.2) is new.

The major difference is that the equations have different fractional order. The order is 3<α≤4 in this paper and 1<α≤2 in [14] and [18]. The higher order leads the comparable process to being more difficult and complex.

Nonlocal boundary conditions are considered in this paper and [18], Dirichlet boundary conditions are considered in [14].

Both the existence and nonexistence are considered in this paper, but only the existence is considered in [14] and [18].

Declarations

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009), also supported by the Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Authors’ Affiliations

(1)

School of Mathematical Sciences, University of Jinan

(2)

Department of Mathematics and Statistics, Missouri University of Science and Technology

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.